ODEs and Wiman-Valiron theory in the unit disc
The order of an analytic function \(f\) in the unit disc is defined by \[ \rho:=\limsup_{r\to 1^{-}}\frac{\log\log M(r,f)}{-\log(1-r)}. \] The authors obtain Wiman-Valiron type theorems: Theorem. Suppose that \(f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\) is analytic in the unit disc, of order \(\rho>0\). Given \(\gamma\) satisfying \[ 0<\gamma<\frac{\rho}{2(\rho+1)}, \] let \(\zeta\) be such that \[ |f(\zeta)|\geq N^{-\gamma}M\big(|\zeta|, f\big), \] where \(N=N\big(|\zeta|,f\big)\). Let \(T_{j}\) be a sequence satisfying \[ \lim_{j\to\infty}T_{j}=1\quad\text{and}\quad \lim_{j\to\infty}\frac{\log N(T_{j}, f)}{-\log(1-T_{j})}=\rho+1. \] Then, for every positive integer \(q\), \[ f^{(q)}(\zeta)=\big(1+o(1)\big)\left(\frac{N}{\zeta}\right)^{q}f(\zeta) \] as \(|\zeta|\to 1^-\) outside a set \(E\) such that \[ \lim_{j\to \infty}\frac{m\big(E\cap(T_{j}, 1)\big)}{1-T_{j}}=0. \] Theorem. Suppose that \(f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\) is analytic in the unit disc, of order zero, and that \(\gamma:(0,1)\to\mathbb{R}\) is positive such that \(\gamma(t)\to 0\) as \(t\to 1^{-}\). If \(\big|f(\zeta)\big|\geq N^{-\gamma(|\zeta|)}M\big(|\zeta|, f\big)\), then for every positive integer \(q\) and any positive number \(\eta\), \[ \frac{f^{(q)}(\zeta)}{f(\zeta)}=O\left(\frac{1}{1-|\zeta|}\right)^{q+\eta} \] as \(|\zeta|\to 1^-\) outside a set of zero final density. The authors also apply the above theorems to ODEs with analytic coefficients in the unit disc.
scientific article; zbMATH DE number 5687687
scientific article